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Rank of a Matrix

The rank of t...

Question

The rank of the matrix $⎡ ⎢ ⎣ \begin{matrix} - 4 & 1 & - 1 - 1 & - 1 & - 1 7 & - 3 & 1 \end{matrix} ⎤ ⎥ ⎦$ is

A

1

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B

2

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C

3

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D

4

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Solution

The correct option is B 2
A = $⎡ ⎢ ⎣ \begin{matrix} - 4 & 1 & - 1 - 1 & - 1 & - 1 7 & - 3 & 1 \end{matrix} ⎤ ⎥ ⎦$
$R_{2} \leftrightarrow R_{1} \to ⎡ ⎢ ⎣ \begin{matrix} - 1 & - 1 & - 1 - 4 & 1 & - 1 7 & - 3 & 1 \end{matrix} ⎤ ⎥ ⎦$

$\begin{matrix} R_{2} \to R_{2} - 4 R_{1} R_{3} \to R_{3} + 7 R_{1} \end{matrix}} T h e n A \sim ⎡ ⎢ ⎣ \begin{matrix} - 1 & - 1 & - 1 0 & 5 & 3 0 & - 10 & - 6 \end{matrix} ⎤ ⎥ ⎦$

$R_{3} \to R_{3} + 2 R_{2} T h e n A \sim ⎡ ⎢ ⎣ \begin{matrix} - 1 & - 1 & - 1 0 & 5 & 3 0 & 0 & 0 \end{matrix} ⎤ ⎥ ⎦$

Hence $ρ (A) = Number of non zero rows = 2$

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